src/HOL/Fun.thy
author webertj
Mon Mar 07 19:30:53 2005 +0100 (2005-03-07)
changeset 15584 3478bb4f93ff
parent 15531 08c8dad8e399
child 15691 900cf45ff0a6
permissions -rw-r--r--
refute_params: default value itself=1 added (for type classes)
     1 (*  Title:      HOL/Fun.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 Notions about functions.
     7 *)
     8 
     9 theory Fun
    10 imports Typedef
    11 begin
    12 
    13 instance set :: (type) order
    14   by (intro_classes,
    15       (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
    16 
    17 constdefs
    18   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
    19    "fun_upd f a b == % x. if x=a then b else f x"
    20 
    21 nonterminals
    22   updbinds updbind
    23 syntax
    24   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
    25   ""         :: "updbind => updbinds"             ("_")
    26   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
    27   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
    28 
    29 translations
    30   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
    31   "f(x:=y)"                     == "fun_upd f x y"
    32 
    33 (* Hint: to define the sum of two functions (or maps), use sum_case.
    34          A nice infix syntax could be defined (in Datatype.thy or below) by
    35 consts
    36   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
    37 translations
    38  "fun_sum" == sum_case
    39 *)
    40 
    41 constdefs
    42  overwrite :: "('a => 'b) => ('a => 'b) => 'a set => ('a => 'b)"
    43               ("_/'(_|/_')"  [900,0,0]900)
    44 "f(g|A) == %a. if a : A then g a else f a"
    45 
    46  id :: "'a => 'a"
    47 "id == %x. x"
    48 
    49  comp :: "['b => 'c, 'a => 'b, 'a] => 'c"   (infixl "o" 55)
    50 "f o g == %x. f(g(x))"
    51 
    52 text{*compatibility*}
    53 lemmas o_def = comp_def
    54 
    55 syntax (xsymbols)
    56   comp :: "['b => 'c, 'a => 'b, 'a] => 'c"        (infixl "\<circ>" 55)
    57 syntax (HTML output)
    58   comp :: "['b => 'c, 'a => 'b, 'a] => 'c"        (infixl "\<circ>" 55)
    59 
    60 
    61 constdefs
    62   inj_on :: "['a => 'b, 'a set] => bool"         (*injective*)
    63     "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
    64 
    65 text{*A common special case: functions injective over the entire domain type.*}
    66 syntax inj   :: "('a => 'b) => bool"
    67 translations
    68   "inj f" == "inj_on f UNIV"
    69 
    70 constdefs
    71   surj :: "('a => 'b) => bool"                   (*surjective*)
    72     "surj f == ! y. ? x. y=f(x)"
    73 
    74   bij :: "('a => 'b) => bool"                    (*bijective*)
    75     "bij f == inj f & surj f"
    76 
    77 
    78 
    79 text{*As a simplification rule, it replaces all function equalities by
    80   first-order equalities.*}
    81 lemma expand_fun_eq: "(f = g) = (! x. f(x)=g(x))"
    82 apply (rule iffI)
    83 apply (simp (no_asm_simp))
    84 apply (rule ext, simp (no_asm_simp))
    85 done
    86 
    87 lemma apply_inverse:
    88     "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)"
    89 by auto
    90 
    91 
    92 text{*The Identity Function: @{term id}*}
    93 lemma id_apply [simp]: "id x = x"
    94 by (simp add: id_def)
    95 
    96 lemma inj_on_id: "inj_on id A"
    97 by (simp add: inj_on_def) 
    98 
    99 lemma surj_id: "surj id"
   100 by (simp add: surj_def) 
   101 
   102 lemma bij_id: "bij id"
   103 by (simp add: bij_def inj_on_id surj_id) 
   104 
   105 
   106 
   107 subsection{*The Composition Operator: @{term "f \<circ> g"}*}
   108 
   109 lemma o_apply [simp]: "(f o g) x = f (g x)"
   110 by (simp add: comp_def)
   111 
   112 lemma o_assoc: "f o (g o h) = f o g o h"
   113 by (simp add: comp_def)
   114 
   115 lemma id_o [simp]: "id o g = g"
   116 by (simp add: comp_def)
   117 
   118 lemma o_id [simp]: "f o id = f"
   119 by (simp add: comp_def)
   120 
   121 lemma image_compose: "(f o g) ` r = f`(g`r)"
   122 by (simp add: comp_def, blast)
   123 
   124 lemma image_eq_UN: "f`A = (UN x:A. {f x})"
   125 by blast
   126 
   127 lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
   128 by (unfold comp_def, blast)
   129 
   130 
   131 subsection{*The Injectivity Predicate, @{term inj}*}
   132 
   133 text{*NB: @{term inj} now just translates to @{term inj_on}*}
   134 
   135 
   136 text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*}
   137 lemma datatype_injI:
   138     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
   139 by (simp add: inj_on_def)
   140 
   141 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
   142   by (unfold inj_on_def, blast)
   143 
   144 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
   145 by (simp add: inj_on_def)
   146 
   147 (*Useful with the simplifier*)
   148 lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
   149 by (force simp add: inj_on_def)
   150 
   151 
   152 subsection{*The Predicate @{term inj_on}: Injectivity On A Restricted Domain*}
   153 
   154 lemma inj_onI:
   155     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
   156 by (simp add: inj_on_def)
   157 
   158 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
   159 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
   160 
   161 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
   162 by (unfold inj_on_def, blast)
   163 
   164 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
   165 by (blast dest!: inj_onD)
   166 
   167 lemma comp_inj_on:
   168      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
   169 by (simp add: comp_def inj_on_def)
   170 
   171 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
   172 apply(simp add:inj_on_def image_def)
   173 apply blast
   174 done
   175 
   176 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
   177   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
   178 apply(unfold inj_on_def)
   179 apply blast
   180 done
   181 
   182 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
   183 by (unfold inj_on_def, blast)
   184 
   185 lemma inj_singleton: "inj (%s. {s})"
   186 by (simp add: inj_on_def)
   187 
   188 lemma inj_on_empty[iff]: "inj_on f {}"
   189 by(simp add: inj_on_def)
   190 
   191 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
   192 by (unfold inj_on_def, blast)
   193 
   194 lemma inj_on_Un:
   195  "inj_on f (A Un B) =
   196   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
   197 apply(unfold inj_on_def)
   198 apply (blast intro:sym)
   199 done
   200 
   201 lemma inj_on_insert[iff]:
   202   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
   203 apply(unfold inj_on_def)
   204 apply (blast intro:sym)
   205 done
   206 
   207 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
   208 apply(unfold inj_on_def)
   209 apply (blast)
   210 done
   211 
   212 
   213 subsection{*The Predicate @{term surj}: Surjectivity*}
   214 
   215 lemma surjI: "(!! x. g(f x) = x) ==> surj g"
   216 apply (simp add: surj_def)
   217 apply (blast intro: sym)
   218 done
   219 
   220 lemma surj_range: "surj f ==> range f = UNIV"
   221 by (auto simp add: surj_def)
   222 
   223 lemma surjD: "surj f ==> EX x. y = f x"
   224 by (simp add: surj_def)
   225 
   226 lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
   227 by (simp add: surj_def, blast)
   228 
   229 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
   230 apply (simp add: comp_def surj_def, clarify)
   231 apply (drule_tac x = y in spec, clarify)
   232 apply (drule_tac x = x in spec, blast)
   233 done
   234 
   235 
   236 
   237 subsection{*The Predicate @{term bij}: Bijectivity*}
   238 
   239 lemma bijI: "[| inj f; surj f |] ==> bij f"
   240 by (simp add: bij_def)
   241 
   242 lemma bij_is_inj: "bij f ==> inj f"
   243 by (simp add: bij_def)
   244 
   245 lemma bij_is_surj: "bij f ==> surj f"
   246 by (simp add: bij_def)
   247 
   248 
   249 subsection{*Facts About the Identity Function*}
   250 
   251 text{*We seem to need both the @{term id} forms and the @{term "\<lambda>x. x"}
   252 forms. The latter can arise by rewriting, while @{term id} may be used
   253 explicitly.*}
   254 
   255 lemma image_ident [simp]: "(%x. x) ` Y = Y"
   256 by blast
   257 
   258 lemma image_id [simp]: "id ` Y = Y"
   259 by (simp add: id_def)
   260 
   261 lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
   262 by blast
   263 
   264 lemma vimage_id [simp]: "id -` A = A"
   265 by (simp add: id_def)
   266 
   267 lemma vimage_image_eq: "f -` (f ` A) = {y. EX x:A. f x = f y}"
   268 by (blast intro: sym)
   269 
   270 lemma image_vimage_subset: "f ` (f -` A) <= A"
   271 by blast
   272 
   273 lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
   274 by blast
   275 
   276 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
   277 by (simp add: surj_range)
   278 
   279 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
   280 by (simp add: inj_on_def, blast)
   281 
   282 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
   283 apply (unfold surj_def)
   284 apply (blast intro: sym)
   285 done
   286 
   287 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
   288 by (unfold inj_on_def, blast)
   289 
   290 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
   291 apply (unfold bij_def)
   292 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
   293 done
   294 
   295 lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
   296 by blast
   297 
   298 lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
   299 by blast
   300 
   301 lemma inj_on_image_Int:
   302    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
   303 apply (simp add: inj_on_def, blast)
   304 done
   305 
   306 lemma inj_on_image_set_diff:
   307    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
   308 apply (simp add: inj_on_def, blast)
   309 done
   310 
   311 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
   312 by (simp add: inj_on_def, blast)
   313 
   314 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
   315 by (simp add: inj_on_def, blast)
   316 
   317 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
   318 by (blast dest: injD)
   319 
   320 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
   321 by (simp add: inj_on_def, blast)
   322 
   323 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
   324 by (blast dest: injD)
   325 
   326 lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
   327 by blast
   328 
   329 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
   330 lemma image_INT:
   331    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
   332     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   333 apply (simp add: inj_on_def, blast)
   334 done
   335 
   336 (*Compare with image_INT: no use of inj_on, and if f is surjective then
   337   it doesn't matter whether A is empty*)
   338 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   339 apply (simp add: bij_def)
   340 apply (simp add: inj_on_def surj_def, blast)
   341 done
   342 
   343 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
   344 by (auto simp add: surj_def)
   345 
   346 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
   347 by (auto simp add: inj_on_def)
   348 
   349 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
   350 apply (simp add: bij_def)
   351 apply (rule equalityI)
   352 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
   353 done
   354 
   355 
   356 subsection{*Function Updating*}
   357 
   358 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
   359 apply (simp add: fun_upd_def, safe)
   360 apply (erule subst)
   361 apply (rule_tac [2] ext, auto)
   362 done
   363 
   364 (* f x = y ==> f(x:=y) = f *)
   365 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
   366 
   367 (* f(x := f x) = f *)
   368 declare refl [THEN fun_upd_idem, iff]
   369 
   370 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
   371 apply (simp (no_asm) add: fun_upd_def)
   372 done
   373 
   374 (* fun_upd_apply supersedes these two,   but they are useful
   375    if fun_upd_apply is intentionally removed from the simpset *)
   376 lemma fun_upd_same: "(f(x:=y)) x = y"
   377 by simp
   378 
   379 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
   380 by simp
   381 
   382 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
   383 by (simp add: expand_fun_eq)
   384 
   385 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
   386 by (rule ext, auto)
   387 
   388 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
   389 by(fastsimp simp:inj_on_def image_def)
   390 
   391 lemma fun_upd_image:
   392      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
   393 by auto
   394 
   395 subsection{* overwrite *}
   396 
   397 lemma overwrite_emptyset[simp]: "f(g|{}) = f"
   398 by(simp add:overwrite_def)
   399 
   400 lemma overwrite_apply_notin[simp]: "a ~: A ==> (f(g|A)) a = f a"
   401 by(simp add:overwrite_def)
   402 
   403 lemma overwrite_apply_in[simp]: "a : A ==> (f(g|A)) a = g a"
   404 by(simp add:overwrite_def)
   405 
   406 subsection{* swap *}
   407 
   408 constdefs
   409   swap :: "['a, 'a, 'a => 'b] => ('a => 'b)"
   410    "swap a b f == f(a := f b, b:= f a)"
   411 
   412 lemma swap_self: "swap a a f = f"
   413 by (simp add: swap_def) 
   414 
   415 lemma swap_commute: "swap a b f = swap b a f"
   416 by (rule ext, simp add: fun_upd_def swap_def)
   417 
   418 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
   419 by (rule ext, simp add: fun_upd_def swap_def)
   420 
   421 lemma inj_on_imp_inj_on_swap:
   422      "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
   423 by (simp add: inj_on_def swap_def, blast)
   424 
   425 lemma inj_on_swap_iff [simp]:
   426   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
   427 proof 
   428   assume "inj_on (swap a b f) A"
   429   with A have "inj_on (swap a b (swap a b f)) A" 
   430     by (rules intro: inj_on_imp_inj_on_swap) 
   431   thus "inj_on f A" by simp 
   432 next
   433   assume "inj_on f A"
   434   with A show "inj_on (swap a b f) A" by (rules intro: inj_on_imp_inj_on_swap)
   435 qed
   436 
   437 lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
   438 apply (simp add: surj_def swap_def, clarify)
   439 apply (rule_tac P = "y = f b" in case_split_thm, blast)
   440 apply (rule_tac P = "y = f a" in case_split_thm, auto)
   441   --{*We don't yet have @{text case_tac}*}
   442 done
   443 
   444 lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
   445 proof 
   446   assume "surj (swap a b f)"
   447   hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
   448   thus "surj f" by simp 
   449 next
   450   assume "surj f"
   451   thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
   452 qed
   453 
   454 lemma bij_swap_iff: "bij (swap a b f) = bij f"
   455 by (simp add: bij_def)
   456  
   457 
   458 text{*The ML section includes some compatibility bindings and a simproc
   459 for function updates, in addition to the usual ML-bindings of theorems.*}
   460 ML
   461 {*
   462 val id_def = thm "id_def";
   463 val inj_on_def = thm "inj_on_def";
   464 val surj_def = thm "surj_def";
   465 val bij_def = thm "bij_def";
   466 val fun_upd_def = thm "fun_upd_def";
   467 
   468 val o_def = thm "comp_def";
   469 val injI = thm "inj_onI";
   470 val inj_inverseI = thm "inj_on_inverseI";
   471 val set_cs = claset() delrules [equalityI];
   472 
   473 val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];
   474 
   475 (* simplifies terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *)
   476 local
   477   fun gen_fun_upd NONE T _ _ = NONE
   478     | gen_fun_upd (SOME f) T x y = SOME (Const ("Fun.fun_upd",T) $ f $ x $ y)
   479   fun dest_fun_T1 (Type (_, T :: Ts)) = T
   480   fun find_double (t as Const ("Fun.fun_upd",T) $ f $ x $ y) =
   481     let
   482       fun find (Const ("Fun.fun_upd",T) $ g $ v $ w) =
   483             if v aconv x then SOME g else gen_fun_upd (find g) T v w
   484         | find t = NONE
   485     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
   486 
   487   val ss = simpset ()
   488   val fun_upd_prover = K (rtac eq_reflection 1 THEN rtac ext 1 THEN simp_tac ss 1)
   489 in
   490   val fun_upd2_simproc =
   491     Simplifier.simproc (Theory.sign_of (the_context ()))
   492       "fun_upd2" ["f(v := w, x := y)"]
   493       (fn sg => fn _ => fn t =>
   494         case find_double t of (T, NONE) => NONE
   495         | (T, SOME rhs) => SOME (Tactic.prove sg [] [] (Term.equals T $ t $ rhs) fun_upd_prover))
   496 end;
   497 Addsimprocs[fun_upd2_simproc];
   498 
   499 val expand_fun_eq = thm "expand_fun_eq";
   500 val apply_inverse = thm "apply_inverse";
   501 val id_apply = thm "id_apply";
   502 val o_apply = thm "o_apply";
   503 val o_assoc = thm "o_assoc";
   504 val id_o = thm "id_o";
   505 val o_id = thm "o_id";
   506 val image_compose = thm "image_compose";
   507 val image_eq_UN = thm "image_eq_UN";
   508 val UN_o = thm "UN_o";
   509 val datatype_injI = thm "datatype_injI";
   510 val injD = thm "injD";
   511 val inj_eq = thm "inj_eq";
   512 val inj_onI = thm "inj_onI";
   513 val inj_on_inverseI = thm "inj_on_inverseI";
   514 val inj_onD = thm "inj_onD";
   515 val inj_on_iff = thm "inj_on_iff";
   516 val comp_inj_on = thm "comp_inj_on";
   517 val inj_on_contraD = thm "inj_on_contraD";
   518 val inj_singleton = thm "inj_singleton";
   519 val subset_inj_on = thm "subset_inj_on";
   520 val surjI = thm "surjI";
   521 val surj_range = thm "surj_range";
   522 val surjD = thm "surjD";
   523 val surjE = thm "surjE";
   524 val comp_surj = thm "comp_surj";
   525 val bijI = thm "bijI";
   526 val bij_is_inj = thm "bij_is_inj";
   527 val bij_is_surj = thm "bij_is_surj";
   528 val image_ident = thm "image_ident";
   529 val image_id = thm "image_id";
   530 val vimage_ident = thm "vimage_ident";
   531 val vimage_id = thm "vimage_id";
   532 val vimage_image_eq = thm "vimage_image_eq";
   533 val image_vimage_subset = thm "image_vimage_subset";
   534 val image_vimage_eq = thm "image_vimage_eq";
   535 val surj_image_vimage_eq = thm "surj_image_vimage_eq";
   536 val inj_vimage_image_eq = thm "inj_vimage_image_eq";
   537 val vimage_subsetD = thm "vimage_subsetD";
   538 val vimage_subsetI = thm "vimage_subsetI";
   539 val vimage_subset_eq = thm "vimage_subset_eq";
   540 val image_Int_subset = thm "image_Int_subset";
   541 val image_diff_subset = thm "image_diff_subset";
   542 val inj_on_image_Int = thm "inj_on_image_Int";
   543 val inj_on_image_set_diff = thm "inj_on_image_set_diff";
   544 val image_Int = thm "image_Int";
   545 val image_set_diff = thm "image_set_diff";
   546 val inj_image_mem_iff = thm "inj_image_mem_iff";
   547 val inj_image_subset_iff = thm "inj_image_subset_iff";
   548 val inj_image_eq_iff = thm "inj_image_eq_iff";
   549 val image_UN = thm "image_UN";
   550 val image_INT = thm "image_INT";
   551 val bij_image_INT = thm "bij_image_INT";
   552 val surj_Compl_image_subset = thm "surj_Compl_image_subset";
   553 val inj_image_Compl_subset = thm "inj_image_Compl_subset";
   554 val bij_image_Compl_eq = thm "bij_image_Compl_eq";
   555 val fun_upd_idem_iff = thm "fun_upd_idem_iff";
   556 val fun_upd_idem = thm "fun_upd_idem";
   557 val fun_upd_apply = thm "fun_upd_apply";
   558 val fun_upd_same = thm "fun_upd_same";
   559 val fun_upd_other = thm "fun_upd_other";
   560 val fun_upd_upd = thm "fun_upd_upd";
   561 val fun_upd_twist = thm "fun_upd_twist";
   562 val range_ex1_eq = thm "range_ex1_eq";
   563 *}
   564 
   565 end